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Integral of 1/(sinx-cosx-2) dx

The solution

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   0                        
   /                        
  |                         
  |            1            
  |   ------------------- dx
  |   sin(x) - cos(x) - 2   
  |                         
 /                          
-2*pi

$$\int\limits_{- 2 \pi}^{0} \frac{1}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) - 2}\, dx$$

Integral(1/(sin(x) - cos(x) - 2), (x, -2*pi, 0))

Detail solution

Rewrite the integrand:

$\frac{1}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) - 2} = - \frac{1}{- \sin{\left(x \right)} + \cos{\left(x \right)} + 2}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{- \sin{\left(x \right)} + \cos{\left(x \right)} + 2}\right)\, dx = - \int \frac{1}{- \sin{\left(x \right)} + \cos{\left(x \right)} + 2}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)$
So, the result is: $- \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)$
Now simplify:

$- \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)}{2} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)$
Add the constant of integration:

$- \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)}{2} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)+ \mathrm{constant}$

The answer is:

$- \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \left(\tan{\left(\frac{x}{2} \right)} - 1\right)}{2} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   /        /x   pi\       /            ___    /x\\\
 |                                    |        |- - --|       |    ___   \/ 2 *tan|-|||
 |          1                     ___ |        |2   2 |       |  \/ 2             \2/||
 | ------------------- dx = C - \/ 2 *|pi*floor|------| + atan|- ----- + ------------||
 | sin(x) - cos(x) - 2                \        \  pi  /       \    2          2      //
 |                                                                                     
/

$$\int \frac{1}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) - 2}\, dx = C - \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      /      /  ___\       \         /          /  ___\\
  ___ |      |\/ 2 |       |     ___ |          |\/ 2 ||
\/ 2 *|- atan|-----| - 2*pi| - \/ 2 *|-pi - atan|-----||
      \      \  2  /       /         \          \  2  //

$$\sqrt{2} \left(- 2 \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right) - \sqrt{2} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)$$

      /      /  ___\       \         /          /  ___\\
  ___ |      |\/ 2 |       |     ___ |          |\/ 2 ||
\/ 2 *|- atan|-----| - 2*pi| - \/ 2 *|-pi - atan|-----||
      \      \  2  /       /         \          \  2  //

$$\sqrt{2} \left(- 2 \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right) - \sqrt{2} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)$$

sqrt(2)*(-atan(sqrt(2)/2) - 2*pi) - sqrt(2)*(-pi - atan(sqrt(2)/2))

Numerical answer [src]

-4.44288293815837

-4.44288293815837

Use the examples entering the upper and lower limits of integration.

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Integral of 1/(sinx-cosx-2) dx

Limits of integration:

The graph:

Piecewise:

The solution